Splitting Algorithms For Maxwell Equations

This paper is mainly concerned with several splitting algorithms of Maxwell equations. Five difference schemes are newly presented:splitting Wendroff scheme, compact splitting scheme, energy conserved scheme in Yee’s grids, conformal multi-symplectic integration and local one dimensional conformal scheme. For the schemes proposed, we analysis the corresponding conver-gence, stability and conservativeness. Numerical experimental confirm our theoretical conclusion.In Chapter 1, we firstly introduce the background of the Maxwell equations. There are many kinds of mathematical expression forms for Maxwell equations, we mainly consider the time-domain differential one. Then a brief introduction of Bridge multi-symplectic structure is given.In Chapter 2, two splitting difference methods:splitting wendroff scheme and high-order compact splitting scheme for two-dimensional Maxwell equations are considered. Both the schemes are unconditional stability, coding simply, economic and energy conservation. we prove that our schemes are energy conserved and unconditional stability and the second one is 6-order accurate in spatial.In Chapter 3, we develop the energy conservational splitting finite-difference-time-domain (EC-S-FDTD) methods for 2-dimensional Maxwell equations surrounding by perfect electric con-ductors in lossy medium. A new energy method is applied to analysis the given schemes and we obtain four energy conservation identities which mean that the our schemes is unconditionally stable under the new discrete modified energy norms.In Chapter 4, The energy conservation properties of the local one-dimensional multisym-plectic (LOD-MS) Preissman scheme is mainly concerned, which is a scheme for solving the 3-dimensional Maxwell equations under the perfectly electric conducting (PEC) boundary condi-tion. Energy analysis method is applied to obtain two energy conservation identities which suggest that the LOD-MS Preissman scheme is unconditionally stable under the new discrete modified en-ergy norms.In Chapter 5, the forced-damped multi-symplectic partial differential equations (PDEs) in 1+m dimensions with their conformal properties are considered. In particular, geometric inte-gration that preserve these conformal conservation laws for 3-D Maxwell’s equations with added dissipation are presented in detail. We will demonstrate that the local one dimensional splitting method preserve global symplecticity. And depending on the theoretical results of the conformal structure and the conformal conservation laws of Maxwell-Hamiltonian system, we finally obtain the local one dimensional conformal Preissman scheme.